Mathbook: Introduction to Sets

Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I’ll put down in two parts on this blog.

It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator’s idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons. Moving forward, I’ll occasionally add a new post here to that effect. The people in my life don’t always understand the math I learned, so this is part of my effort in showing it.

Gabriel’s Horn

Here is one of my favorite “paradoxes” in mathematics, that many people will learn in a first year calculus course. It is called “Gabriel’s Horn.”

First, we take the function $f(x) = \dfrac{1}{x}$. This is a curve that we can imagine beginning at the point $(1,1)$, then quickly sloping down towards the $x$-axis at which point it becomes nearly horizontal. A plot is shown below.

Now imagine taking that curve and rotating it around the $x$-axis, forming an infinitely long shape that looks like the bell of a trumpet, or horn. After doing so, we get a shape that looks roughly like this:

There are two things we wish to determine about this new three-dimensional figure: its volume, and its surface area. We can think of the volume simply as how much stuff we need to fill it completely, while the surface area is how much paint we need to coat the surface. As we will show, this figure has finite volume but infinite surface area. What this means is that we would need an infinite amount of paint in order to just cover the inside surface with a layer of paint. However, if we really wanted to paint the inside, we could take our finite amount of paint and just “pour it in” the horn. We could fill up this infinite horn with a finite amount of paint, and thus end up painting the inside.

The reason this becomes fun is because this is an infinite object. The skinny part of the bell goes on forever, toward $+\infty$. But this is no obstacle for some tools in calculus. The main idea of calculus is to split your object into arbitrarily thin or short pieces. In our case, we will take an arbitrary vertical slice out of our horn at some horizontal point we’ll call $x_1$. The disk we get as a result will be circular, and have some radius $r$. Consider the figure below.

The radius is the distance from the $x$-axis to the curve $f(x) = \dfrac{1}{x}$. But by definition, this is just the $y$-coordinate, $f(x_1) = \dfrac{1}{x_1}$. So, for any arbitrary vertical slice taken out of our horn, we can find the radius. This allows us to easily find the circumference ($2\pi r$) and area ($\pi r^2$) of each disk.

At this point, we’ll have to assume some knowledge of calculus. Consider taking every possible arbitrarily thin disk in the horn, infinitely many of them! If we add their areas over and over again, $\pi r^2 = \pi \left(\dfrac{1}{x}\right)^2$, multiplied by their tiny piece of width we call $dx$, we will obtain the volume of the horn overall. We can do similarly with the circumference, multiplying each tiny bit of circumference, $2 \pi r = 2\pi \dfrac{1}{x}$, with that same bit of width $dx$. This will give us our surface area.

We do this formally by using integration. In particular, we use an “improper” integral, and technically have to take a limit (of course, once I finished calculus I got too lazy to write out those steps normally, but I will now for completeness.)

For volume, our integral is

$\displaystyle \text{Volume} = \pi \int\limits_1^\infty \dfrac{1}{x^2}dx = \pi \lim\limits_{b\to\infty} \int\limits_1^b \dfrac{1}{x^2}dx$

We can evaluate this integral directly, getting

$\displaystyle \text{Volume} = \pi \lim\limits_{b\to\infty}\left. -\dfrac{1}{x} \right|_1^b = \pi \lim\limits_{b\to\infty} \left( -\dfrac{1}{b} + 1\right) = \boxed{\pi}.$

That $-\dfrac{1}{b}$ goes to $0$, so that term disappears. As we stated earlier, the volume is finite. Choose your units, and the volume of that infinite horn above is just $\pi$. Neat!

We can do similarly for the surface area, but our integral will diverge.

$\displaystyle \text{Surface Area} = 2\pi\int\limits_1^\infty \dfrac{1}{x}dx = 2\pi \lim\limits_{b\to\infty} \int_1^b \dfrac{1}{x}dx = 2\pi \lim\limits_{b\to\infty} \left(\ln b - \ln 1\right) = \boxed{\infty}.$

Since $\ln 1= 0$, we just get $\ln b$ going off to infinity, hence the surface area is infinite.

There are many cool problems like this, some with significantly less background necessary to formulate. I’m hoping to mix these in with some other types of posts.

Card Game Simulation

I had another busy week, so I’m taking advantage of old stuff I can recycle.

A month or two ago, I was playing a Solitaire variation my parents taught me when I was younger, and I realized that it was a completely deterministic game once the deck was shuffled. That is, unlike traditional solitaire, there was no element of choice by the player. As such, it made it very easy to write a simulation of it and analyze the details.

The very brief report I wrote up is here, and the simulation code (which is also linked in the report) is here.

The short version, is that it is a break-even game on average, which is pretty interesting. Furthermore, the overall result is normally distributed around breaking even.

I’m trying to include a more well-rounded amount of content here, since math is still very close to my heart and I’d like to only maintain one sight for everything. It will continue to be a mix of things, so that we’re all on the same page.

Suppose you are at a pizza restaurant with your friends. You all agree you want to buy pizza to maximize your pizza-per-dollar. There’s an easy way to make comparisons between pizza sizes and figure out what the best deal is.

My Senior Thesis

Just to have this out for people to look at, here is my honors thesis from my degree. It is broadly about some of the mathematics behind a particular phenomenon in quantum mechanics.

Thesis

Thesis Formatted as a Book

Fun Coin Flips

I learned an interesting fact in my Stochastic Processes class the other day, and I managed to come up with an easier way to present it than using Markov chains (which are really cool, but not conducive to making a good blog post).

Monty Hall and Gambler’s Ruin: A Third Small Step into Mathematics

Today we will be looking at two problems in probability: the Monty Hall problem and Gambler’s Ruin. These are two common probability “brain teasers”. For the Monty Hall problem, it feels paradoxical when you first learn about it, while the Gambler’s Ruin is instructive and important as you go forward in life. With that brief introduction, let us begin!

There was a game show (so I have been told, though I am too young to have watched it) hosted by Monty Hall. One of the main elements of the show was a classic three-door setup, where behind two doors there were goats, and behind the third door there was a brand new car waiting to be won. The way this would go is that Monty Hall would ask you, the contestant, what door you wished to select. You picked door one, two or three. Then Monty Hall, bold and clever, would open up one of the two doors that you did not pick, only to reveal one of the goats! At this point in the game he would come back to you with a smirk, asking if you wished to stay with the door you picked at first, or if you wanted to switch to the other closed door. The problem is: in order to maximize the probability that you win the car, do you switch or stay (or does it even matter)?
Continue reading “Monty Hall and Gambler’s Ruin: A Third Small Step into Mathematics”

Mathmagic Land: A Second Small Step into Mathematics﻿

Mathematics is the alphabet with which God has written the universe.
-Galileo

In our second small step into mathematics, I wish to explore that wonderful world that many generations of students will hopefully be familiar with: Mathmagic Land. This was an Academy Award nominated short film featuring Donald Duck, adventuring through a place entirely unfamiliar to him. After first asserting that math is for “eggheads”, a friendly narrator guides him through the math involved in music, architecture, art and games. In my first post I said I wished to go beyond the utility of math, and I believe this movie transcends utility to explore the natural wonder of math and where it appears in the wild.

This video is just under half an hour and is freely available on YouTube (linked above) so I recommend that you watch it as a prerequisite to this post. I hope to extend some of the concepts that are so nicely presented in the video, while also adding some of the places where I love seeing math show up in organic ways. So without further delay, let us begin!
Continue reading “Mathmagic Land: A Second Small Step into Mathematics﻿”

“Oh, You’re Studying Math?” A First Small Step into Mathematics

“That sounds hard. I used to like math, but then I had a bad teacher and realized I just wasn’t any good at it anymore. I think it’s really cool that you like it though, and that you can teach other people about it. That’s what you want to do, right? Become a math teacher? That sounds really great. I couldn’t be a teacher, especially for math, that sounds way too difficult. Continue reading ““Oh, You’re Studying Math?” A First Small Step into Mathematics”